Convective Mass Flows I

1
Convective Mass Flows I

• In this lecture, we shall begin looking at the
  problem of convective mass flows.
• Irreversible thermodynamics concerns itself with
  purely thermal phenomena, as well as the
  conversion of free energy into heat.
• In reversible thermodynamics, the situation is
  complicated by the fact that it often concerns
  itself with mass flows next to energy flows.

2
Table of Contents

• Mass flow vs. entropy flow
• Fluid moving through a pipe
• Wave equation
• Forced flow
• Turbines
• Compressors and pumps
• Heat flow
• Mass transport losses

3
Mass Flows vs. Entropy Flows

• Although there exist phenomena that take place
  purely in the thermal domain, there can be no
  mass flows that occur without heat flows
  accompanying them.
• The problem has to do with the fact that mass
  flows always carry their volume and stored heat
  with them. It is therefore not meaningful to
  consider these quantities independently of each
  other.
• The water circulation within Biosphere 2 may
  serve as an excellent example. The thermal
  phenomena of the water cycle cannot be properly
  described without taking into account its mass
  flow (or at least its volume flow) as well.

4
Fluid Moving in a Pipe I

• We shall start by modeling the flow of fluids
  (liquids or gases) in a pipe.
• The pipe can be subdivided into segments of
  length Dx.
• If more fluid enters a segment than leaves it,
  the pressure of the fluid in the segment must
  evidently grow. The fluid is being compressed.

5
Fluid Moving in a Pipe II

• When the pressure at the entrance of a segment is
  higher that at the exit, the speed of the fluid
  must increase. This effect is caused by the
  mechanical nature of the fluid. The pressure is
  proportional to a force, and the volume is
  proportional to the mass. Consequently, this is
  an inductive phenomenon. It describes the
  inertia of the moving mass.

6
Fluid Moving in a Pipe III

• Consequently, the following bond graph may be
  proposed

7
Capacitors and Inductors

• Although the hydraulic/pneumatic inductor
  describes the same physical phenomenon as the
  mechanical inductor, its measurement units are
  nevertheless different.

8
The Wave Equation I

• Any text book of physics teaches us that moving
  fluid in a pipe satisfies the wave equation.
• Discretization in space leads to

9
The Wave Equation II

• The following replacement circuit comes to mind

10
The Wave Equation III

• A chain of such links indeed corresponds to the
  proposed bond graph

11
The Forced Flow I

• A forced flow can first be conceptualized as a
  modulated flow source.

What happens with the energy at the interrupted
chain? As the flows to the left and to the right
are identical (mass conservation), it makes sense
to unite the two flow sources.
12
The Forced Flow II
It seems that we need to look at this problem a
little more carefully ...
13
The Turbine I

• We shall check what happens, when the blade of a
  turbine is placed in the path of a flow that
  occurs for reasons external to the model.
• The pressure difference on the two sides of the
  blade generates a resulting force that produces a
  torque in the turbine.
• The generated torque is proportional to the
  pressure difference, and consequently, we
  recognize the effects of a bond-graphic
  transformer at work.
• If the turbine is designed optimally, the
  pressure difference is fully converted to a
  torque, i.e., no hydraulic energy is left to be
  stored in an inductor.

14
The Turbine II
The pressure difference ?pi leads to a torque ?.
This in turn generates an angular velocity ? at
the turbine, which induces a flow qi back on the
hydraulic side.
15
Compressors and Pumps I

• Cause and effect can also be reversed. We can
  generate a torque at the turbine by means of a
  DC-motor sitting on the same axle. The turbine
  together with the motor is now called either
  compressor or pump. This device induces a flow
  qi on the hydraulic side, which causes a
  corresponding pressure difference.

16
Compressors and Pumps II
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qi
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Dpi
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17
Heat Flow I

• We have meanwhile understood, how the mass flow
  needs to be modeled. However, the transported
  mass always carries its own heat along.
• When modeling thermal phenomena, it is therefore
  important, to correctly represent these heat
  flows, which are not of a dissipative nature.

18
Heat Flow II

• The resulting heat flow can indeed be represented
  as a non-linear (modulated) flow source.

19
Mass Transport Losses

• Fluid transports are in reality always associated
  with losses due to friction.

20
Conclusions

• We have meanwhile understood that the cause of
  convective heat flows is to be found in the mass
  transport.
• The mass transport is captured by the wave
  equation, whereby forced flows (pumps,
  compressors) replace the inductors at the
  locations of flow-forcing devices.
• The heat flow is a consequence of the mass flow,
  and may be modeled using non-linear (modulated)
  internal flow sources.
• Frictional losses can be added to the model
  afterwards where and when needed.